CMO Tranches Introduction - Leeds School of Business

CMO Tranches Introduction - Leeds School of Business

CMOs, IOs, POs and Valuation CMO Mechanics Interest Only Principal Only CMO Tranches: PACs, TACs, Floaters and Inverse Floaters MBS Valuation o MBS Prices o Static Valuation Models Static Yield Spread Spread to the Yield Curve o Dynamic Valuation Models Option Adjusted Spreads o Introduction to Interest Rate Modeling CMOs Introduced by Freddie Mac in June 1983 Created with separate classes or tranches where investors have distinct claims to MBS cash flows Typically have at least four tranches Prepayment risk not shared equally Institutional maturity intermediation Most have AAA ratings CMOs Class A (Fast Pay) Bondholders receive o Coupon interest o All scheduled principal payment (amortization) o All unscheduled principal repayment (prepayments) o Interest that accrues to Zero Coupon Tranche is reclassified principal and paid to Class A bondholders While Class A bondholders are being repaid, Class B and C bondholders receive coupon interest only. Zero Coupon bonds accrue interest until other tranches retired CMOs CMO Residuals Income from securitized mortgages exceed bond payments and expenses because o Overcollateralization o Bond Coupons < Mortgage Interest Rates Residual typically held by issuer CMO residual yields typically 300-500bp above other mortgage derivative products Loan Amount for One Loan Annual Interest Rate $100,000.00 8.00% Loan Term in Years 30 Number of Loans in Pool 1,000

Fees: Servicing Fee (basis points) Guarantee Fee (basis points) Prepay Rate: % of PSA 44 6 100.00% Computed: Monthly Payment for One Loan Number of Payments Assets Pledged as Security Debt Issued against Pool Equity Residual IRR $733.76 360 100,000,000 97,000,000 3,000,000 13.56% Tranche Coupon Rate Amount Issued Weight Weighted Average Coupon A 6.00% $30,000,000 0.3093 1.86% B 6.50% $25,000,000 0.2577 1.68% C 7.00% $20,000,000

0.2062 1.44% Z 8.00% $22,000,000 0.2268 1.81% $97,000,000 1.0000 6.79% Total Prepayment Rate Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Number of Prepayments 12 36 54 52 49

47 44 41 39 36 35 32 31 28 28 25 24 23 21 20 19 18 16 16 15 14 13 12 12 11 = Residual IRR = 100 % of PSA MBS Cash In $9,967,310 $12,117,255 $13,446,880 $12,724,947 $11,933,158 $11,262,832 $10,522,280 $9,809,618 $9,213,849 $8,553,493 $8,088,003 $7,469,514 $7,042,820 $6,469,509 $6,155,155 $5,621,021 $5,260,113 $4,908,755 $4,505,546 $4,184,354 $3,874,822 $3,576,811 $3,242,892 $3,020,416 $2,758,150 $2,508,870 $2,273,357 $2,053,083 $1,857,285 $1,654,139

Fees $496,374 $480,379 $453,813 $423,043 $393,908 $366,320 $340,148 $315,458 $292,158 $270,227 $249,284 $229,464 $210,819 $193,088 $176,357 $160,381 $145,365 $130,984 $117,402 $104,457 $92,161 $80,469 $69,316 $58,706 $48,577 $38,913 $29,675 $20,858 $12,395 $4,267 Tranche A $5,558,313 $7,810,127 $9,287,049 $8,732,636 $3,180,383 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0

MBS Cash Out Tranche B $1,625,000 $1,625,000 $1,625,000 $1,625,000 $6,543,399 $9,178,955 $8,559,256 $3,917,566 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Tranche C $1,400,000 $1,400,000 $1,400,000 $1,400,000 $1,400,000 $1,400,000 $1,400,000 $5,444,462 $8,851,484 $8,196,380 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0

$0 Tranche Z $0 $0 $0 $0 $0 $0 $0 $0 $0 $72,065 $7,838,719 $7,240,050 $6,832,001 $6,276,422 $5,978,797 $5,460,640 $5,114,748 $4,777,771 $4,388,144 $4,079,897 $3,782,661 $3,496,342 $3,173,576 $2,961,710 $2,684,112 $2,266,373 $1,305,268 $0 $0 $0 13.56% Residual $887,624 $801,750 $681,018 $544,268 $415,467 $317,558 $222,876 $132,132 $70,207 $14,820 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $25,461 $203,584 $938,414

$2,032,224 $1,844,890 $1,649,873 Prepayment Rate Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Number of Prepayments 25 70 99 91 82 71 64 57 50 44 40 35 31 27 24 22 19 17 15 13 12 10 10 8

7 7 5 5 5 4 = Residual IRR = 200 % of PSA MBS Cash In $11,225,966 $15,235,852 $17,249,869 $15,523,777 $13,798,570 $11,984,479 $10,641,439 $9,377,650 $8,186,882 $7,155,438 $6,359,801 $5,528,949 $4,833,837 $4,184,363 $3,656,075 $3,239,362 $2,777,331 $2,420,671 $2,089,736 $1,785,891 $1,566,621 $1,307,697 $1,174,218 $956,240 $806,390 $703,875 $552,349 $473,166 $392,010 $307,743 Fees $494,468 $467,224 $420,777 $369,537 $323,824 $283,442 $247,836 $216,419 $188,617 $164,133 $142,598 $123,580 $106,874 $92,173 $79,216 $67,879 $57,825 $49,095

$41,398 $34,690 $28,890 $23,661 $19,225 $15,315 $11,920 $8,998 $6,452 $4,276 $2,393 $781 Tranche A $6,824,589 $10,981,344 $13,222,180 $2,484,055 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 MBS Cash Out Tranche B $1,625,000 $1,625,000 $1,625,000 $10,869,218 $11,803,090 $4,475,946 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0

$0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Tranche C $1,400,000 $1,400,000 $1,400,000 $1,400,000 $1,400,000 $7,066,065 $10,304,047 $4,703,978 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Tranche Z $0 $0 $0 $0 $0 $0 $0 $4,435,811 $7,998,265 $6,991,305 $6,217,203 $5,405,369 $4,726,963 $4,092,190 $3,576,860 $3,171,483

$2,719,507 $2,370,701 $1,946,794 $1,503,449 $1,027,616 $0 $0 $0 $0 $0 $0 $0 $0 $0 11.38% Residual $881,908 $762,285 $581,911 $400,966 $271,656 $159,026 $89,556 $21,441 $0 $0 $0 $0 $0 $0 $0 $0 $0 $875 $101,544 $247,751 $510,114 $1,284,036 $1,154,992 $940,926 $794,470 $694,877 $545,896 $468,890 $389,618 $306,962 Prepayment Rate Year 1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Number of Prepayments 50 132 169 140 110 86 67 53 42 32 26 20 16 12 10 7 6 5 4 2 3 2 1 1 1 1 0 1 0 0 = Residual IRR = 400 % of PSA MBS Cash In $13,642,784 $20,900,811 $23,032,404 $18,666,718 $14,539,056

$11,274,860 $8,714,605 $6,804,439 $5,312,897 $4,029,561 $3,193,882 $2,432,957 $1,901,934 $1,418,596 $1,135,543 $806,816 $655,390 $518,838 $398,054 $230,486 $258,976 $171,759 $98,951 $84,139 $70,154 $56,268 $17,610 $33,835 $8,805 $8,806 Fees $490,575 $441,940 $361,912 $281,712 $218,527 $169,331 $131,046 $101,338 $78,215 $60,240 $46,287 $35,560 $27,261 $20,772 $15,809 $12,000 $9,091 $6,789 $5,080 $3,758 $2,789 $2,018 $1,432 $1,009 $707 $489 $271 $202 $64 $23 Tranche A $9,256,982 $16,747,438 $6,657,899 $0 $0

$0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 MBS Cash Out Tranche B $1,625,000 $1,625,000 $14,188,266 $12,501,125 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Tranche C $1,400,000 $1,400,000 $1,400,000

$5,658,137 $14,205,995 $2,176,753 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Tranche Z $0 $0 $0 $0 $0 $8,917,308 $8,583,559 $6,703,101 $5,234,682 $3,969,321 $3,147,594 $2,333,606 $1,647,412 $304,582 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 10.89% Residual $870,227

$686,433 $424,326 $225,744 $114,533 $11,468 $0 $0 $0 $0 $0 $63,791 $227,261 $1,093,242 $1,119,734 $794,816 $646,299 $512,049 $392,973 $226,728 $256,188 $169,741 $97,519 $83,130 $69,447 $55,780 $17,340 $33,633 $8,742 $8,782 Prepayment Rate Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of Prepayments 6 18 28 28 27 27 25 25 24 24 22 22 22 21 20 19 19 19 18 17 17 16 16 15 15 15 14 14 13 13 = Residual IRR = 50 % of PSA MBS Cash In $9,388,229 $10,463,189 $11,219,397 $10,942,251 $10,572,151 $10,303,908 $9,851,699 $9,594,515 $9,252,289 $9,001,329 $8,580,569 $8,342,000 $8,104,413 $7,786,262 $7,476,005 $7,173,584 $6,949,536 $6,724,347 $6,433,378 $6,149,600 $5,932,416 $5,659,181 $5,443,159 $5,181,342

$4,967,210 $4,747,252 $4,497,197 $4,277,007 $4,040,401 $3,820,206 Fees $497,368 $487,203 $471,206 $452,637 $434,251 $416,202 $398,281 $380,716 $363,488 $346,248 $329,211 $312,436 $295,798 $279,180 $262,707 $246,326 $229,956 $213,614 $197,308 $180,908 $164,528 $148,018 $131,408 $114,693 $97,824 $80,742 $63,442 $45,903 $28,074 $9,950 Tranche A $4,975,255 $6,128,765 $6,989,994 $6,831,564 $6,583,133 $4,212,187 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0

$0 $0 $0 $0 $0 $0 MBS Cash Out Tranche B $1,625,000 $1,625,000 $1,625,000 $1,625,000 $1,625,000 $3,850,583 $7,714,274 $7,555,638 $7,314,752 $2,786,676 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Tranche C $1,400,000 $1,400,000 $1,400,000 $1,400,000 $1,400,000 $1,400,000 $1,400,000 $1,400,000 $1,400,000 $5,776,702 $8,218,776 $8,029,443 $471,454 $0 $0 $0 $0 $0 $0 $0 $0 $0

$0 $0 $0 $0 $0 $0 $0 $0 Tranche Z $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $7,337,161 $7,507,082 $7,213,299 $6,927,258 $6,719,579 $6,510,733 $6,236,070 $5,968,691 $5,767,888 $5,511,163 $5,311,751 $5,066,649 $4,869,386 $4,666,510 $4,433,755 $4,228,829 $3,150,093 $0 16.73% Residual $890,606 $822,222 $733,197 $633,051 $529,766 $424,935 $339,143 $258,161 $174,049 $91,703 $32,581 $120 $0 $0 $0 $0 $0 $0 $0 $0

$0 $0 $0 $0 $0 $0 $0 $2,275 $862,235 $3,810,256 CMOs Interest Only IO Strip: the interest only portion of a CMO (tranche) Investor benefits from slowing prepayments--interest payments made for longer than expected period when prepayments are slower than expected Bearish security: prices tend to rise when interest rates rise CMOs Principal Only PO Strip: principal only strip purchased at a discount If prepayments are faster than expected (e.g. in a declining interest rate environment), then investors receive CFs sooner than expected increasing the investors' yield Prepayment Rate Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Number of Prepayments 12 36

54 52 49 47 44 41 39 36 35 32 31 28 28 25 24 23 21 20 19 18 16 16 15 14 13 12 12 11 = 100 % MBS Cash In $9,967,310 $12,117,255 $13,446,880 $12,724,947 $11,933,158 $11,262,832 $10,522,280 $9,809,618 $9,213,849 $8,553,493 $8,088,003 $7,469,514 $7,042,820 $6,469,509 $6,155,155 $5,621,021 $5,260,113 $4,908,755 $4,505,546 $4,184,354 $3,874,822 $3,576,811 $3,242,892 $3,020,416 $2,758,150 $2,508,870 $2,273,357 $2,053,083 $1,857,285 $1,654,139

MBS Cash Out Principal Interest $2,025,331 $7,445,606 $4,431,197 $7,205,680 $6,185,875 $6,807,192 $5,956,264 $6,345,640 $5,630,627 $5,908,623 $5,401,709 $5,494,803 $5,079,914 $5,102,218 $4,762,286 $4,731,874 $4,539,326 $4,382,365 $4,229,857 $4,053,409 $4,099,456 $3,739,263 $3,798,097 $3,441,954 $3,669,715 $3,162,285 $3,380,103 $2,896,319 $3,333,437 $2,645,360 $3,054,927 $2,405,713 $2,934,273 $2,180,475 $2,813,014 $1,964,757 $2,627,119 $1,761,025 $2,513,043 $1,566,855 $2,400,243 $1,382,417 $2,289,302 $1,207,040 $2,133,831 $1,039,745 $2,081,128 $880,583 $1,980,916 $728,657 $1,886,268 $583,689 $1,798,551 $445,131 $1,719,351 $312,874 $1,658,967 $185,923 $1,585,874 $63,999

Fees $496,374 $480,379 $453,813 $423,043 $393,908 $366,320 $340,148 $315,458 $292,158 $270,227 $249,284 $229,464 $210,819 $193,088 $176,357 $160,381 $145,365 $130,984 $117,402 $104,457 $92,161 $80,469 $69,316 $58,706 $48,577 $38,913 $29,675 $20,858 $12,395 $4,267 Principal Only Interest Only Discount Rate 8.25% 7.75% Price @ 100 PSA $44,922,302.05 $51,558,714.96 Yields: @ 50 PSA @ 100 PSA @ 200 PSA @ 400 PSA 6.08% 8.25% 12.64% 20.12% 10.20% 7.75% 2.88% -6.72%

CMO Tranches Introduction In addition to the sequential pay tranches that include a zero-coupon (or accretion) bond, collateralized mortgage obligations typically have one (or more) of the following tranches: PAC--planned amortization class TAC--targeted amortization class Floating rate tranche Inverse floating rate tranche CMO Tranches Planned Amortization Classes Introduced in 1986 Prepayment "protected" CMO tranche Bond pays scheduled principal payments (irrespective of CFs into the CMO) for a range of actual prepayment speeds Range of prepay rates: 75% PSA to 240% PSA A companion PAC bond absorbs any cash flow uncertainty o PAC and companion PAC receive coupon interest, but o All CMO principal in excess of PAC scheduled go to companion o All CMO principal below PAC scheduled paid from companion CMO Tranches Targeted Amortization Classes CMO pricing speed = prepayment rate that was assumed when the issue was priced A TAC is a PAC with o A lower band = CMO pricing speed o An upper band similar to a PAC upper band Like PACs, TACs repay principal according to a schedule as long as actual prepayments are within a specified range Unlike PACs, TACs DO NOT protect against WAL extensions when prepayments are slower than pricing speed Investors tradeoff extension risk for higher yields CMO Tranches Floaters and Inverse Floaters Introduced by Shearson Lehman Brothers in 1986 Divide a fixed rate CMO tranche into two tranches with variable coupons o Floater rate moves with some index rate o Inverse floater varies inversely with the index Collateral weighted average coupon = fixed rate coupon Principal payments are the same as they are for the underlying fixed rate tranche CMO Tranches Floating Rate Tranche Floating rate = Index + spread (in basis points) Also have maximums, or caps, on floating rate Typical index rates include o LIBOR: London Interbank Overnight Rate o COFI: FHLB 11th District Cost of Funds Index

o CMT: Constant Maturity Treasuries Example: Floating rate = LIBOR + 65bp with cap of 12% CMO Tranches Inverse Floater C IFF Inverse Floater w h e re C = th e c o u p o n o n th e fix e d ra te c o lla te ra l IFF Multiplier Coupon Inverse floater ( Index factor Floater Inverse face floater Spread) Inverse floater Collateral value face value In v e rs e flo a te r tra n c h e s h a v e in te re s t ra te flo o rs . Multiplier face face value value

, CMO Tranches Inverse Floater E x a m p l e : S u p p o s e a $ 3 0 M 6 % f ix e d r a te c o u p o n C M O tr a n c h e w a s d iv id e d in to a $ 2 0 M f lo a tin g r a te b o n d a n d a $ 1 0 M in v e r s e f lo a te r . If th e c o u p o n o n th e f lo a ti n g r a te b o n d is L IB O R + 5 0 b p , w h a t is th e c o u p o n o n th e in v e r s e f lo a te r ? M u ltip lie r = $ 2 0 M /$ 1 0 M = 2 ; Inverse Floater Coupon IF F = 1 0 /3 0 = 1 /3 6 ( LIBOR 1 3 0.50) 18 - 2 LIBOR - 1.00 17.0 - 2 LIBOR 2 CMO Tranches Inverse Floater Note that the (collateral) weighted coupon is: $ 20 M $10M ( LIBOR 0.5) (17 .0 2 LIBOR) $ 30 M $30M 2 1 1 2 LIBOR 17 .0 LIBOR 3 3 3 3 18 6 %, the coupon on the fixed rate tranche 3 Valuation MBS Prices Stated as a percentage of the face amount (e.g. 102) Fractions of one percent are expressed in thirty-seconds (1/32)

Sixty fourths (1/64) are represented with a + Valuation Ginnie Mae 6.5; WAC 7.0% Price 100-24, Assumed WAM 29-08 Prepayment Speed Yield at Projected Speed WAL @ Projected Speed Yield of WAL (9.4yr) Treasury Spread over Treasury 125 PSA 6.42% 9.4 Years 4.90% 152bp Valuation Scenario Analysis for Ginnie Mae 6.5% Pass Through (Short-Term ) Interest Rate Moves (bp) -100 Prepay rate (PSA) WAL (years) Yield WAL Treasury Yield Spread/WAL (bp) Yield Curve 1 Yr 4.63 0 425 3.2 6.19% 3.78% 241 2 Yr 4.77 Source: Salomon Smith Barney 3 Yr 4.78 125 9.4 6.43% 4.90% 142 5 Yr 4.79 +100 112 10.4 10.4% 5.91% 53 10 Yr 4.91 30 Yr

5.35 Valuation Things Influencing MBS Cash Flows Aging: newer MBSs tend to have slower prepay speeds than seasoned MBSs because households that just moved are unlikely to change homes soon after moving Burnout: when market interest rates fall below coupon, prepay speeds accelerate rapidly. After the most financially sophisticated borrowers exit the pool, prepay speeds slow down significantly (burnout) Seasoning: home buying (selling) has a seasonal component with more transactions (and prepayments) in the summer months and fewer transactions (and prepayments) in the winter months. Valuation Valuation Models 1. Static yield spread (Spread/WAL): the spread between the MBS bond yield and the yield on the benchmark Treasury. 2. Yield curve spread: discount each monthly cash flow from the MBS using the yield on the same maturity Treasury plus a constant spread a. Spread to Treasury zero b. Spread to forward rate 3. Option Adjusted Spread: use Monte Carlo methods to simulate alternative interest rate paths; associate expected prepayments; take average of resulting PVs Valuation Yield Curve Spread For a given yield curve (or series of zero Treasury rates), rt, the PV of the MBS cash flows is: PV(s) CF3 CF1 CF2 CFn .... 1 r1 s (1 r2 s) 2 (1 r3 s) 3 (1 rn s) n where CFi is the ith period cash flow, ri is the ith period zero rate, and s is the spread to the yield curve. The value of s that equates PV(s) to the market price of the MBS is the Yield Curve Spread. Valuation Option Adjusted Spread The Yield Curve Spread (and the Spread/WAL) ignore uncertainty in future interest rates (e.g. they use the current yield on Treasuries to value MBSs) Future prepayments are going to depend on future interest ratesthese are likely to be different from current rates In addition, investors reinvestment rates will depend on future interest rates, not on the current yield curve. Valuation Option Adjusted Spread

1. Use some interest rate model to generate an interest rate series 2. Attach prepayment behavior to the generated interest rates 3. Compute the expected MBS cash flows for the assumed interest rate/prepayment behavior. 4. Use the yield curve spread to compute PV(s) for the assumed interest rate series. 5. Repeat 1-4 one thousand times. 6. The value of s that equates the average PV(s) to the current MBS price is the Option Adjusted Spread. Valuation OASs for Ginnie Mae Pass-Throughs Coupon WAM Price PSA Yield WAL Spread/ YCS OAS WAL 6.0% 6.5% 7.0% 7.5% 8.0% 8.5% 29-10 29-08 28-04 28-02 28-01 28-00 98-17 100-23 102-10 103-10 104-10 106-02 85 125 210 340 390 385 6.24 11.4yrs 130bp 6.42 9.5 153 6.51 6.5 166 6.42 4.2 160 6.41 3.6 160 6.33 3.7 152 Source: Salomon Smith Barney 110bp 134 153 161 167 158 70bp

75 78 85 105 108 Valuation Introduction to Interest Rate Modeling Interest Rate Models: 1. Must be consistent with the current yield curvebenchmark Treasuries must be fairly priced 2. Cannot lead to arbitrage opportunities 3. Must be consistent with historical experience a. No negative rates b. Cannot allow rates to increase without bound c. Must incorporate relative volatility in the TERM STRUCTURE: historically, yields on 1-Yr Treasuries are more volatile then either 1-Mo rates or 10-Yr rates. Valuation TREASURY YIELD CURVE RATES FOR JUN 2004 Date 06/01/2004 06/02/2004 06/03/2004 06/04/2004 06/07/2004 06/08/2004 06/09/2004 06/10/2004 06/14/2004 06/15/2004 06/16/2004 06/17/2004 06/18/2004 06/21/2004 06/22/2004 1-mo 0.97 0.97 0.96 0.96 0.97 1.03 1.03 1.02 1.07 1.09 1.05 1.02 1.03 1.04 1.08 3-mo 1.17 1.17 1.17 1.21 1.24 1.27 1.27 1.30

1.41 1.34 1.30 1.28 1.29 1.33 1.32 6-mo 1.44 1.45 1.45 1.51 1.54 1.56 1.61 1.65 1.77 1.68 1.69 1.66 1.68 1.70 1.69 1-yr 1.89 1.92 1.91 1.97 1.96 2.02 2.14 2.14 2.25 2.16 2.24 2.21 2.24 2.17 2.21 The 30-yr has been discontinued. 2-yr 2.60 2.65 2.63 2.70 2.67 2.73 2.79 2.81 2.97 2.77 2.84 2.81 2.81 2.80 2.80 3-yr 3.14 3.19

3.17 3.25 3.22 3.24 3.31 3.32 3.45 3.26 3.31 3.28 3.28 3.26 3.28 5-yr 3.86 3.91 3.89 3.97 3.95 3.96 4.01 4.00 4.10 3.90 3.96 3.93 3.94 3.91 3.92 7-yr 10-yr 20-yr 4.31 4.71 5.45 4.35 4.74 5.47 4.34 4.71 5.46 4.41 4.78 5.51 4.39 4.78 5.51 4.40 4.78 5.50 4.44 4.82 5.54 4.43 4.81 5.52 4.51 4.89 5.59 4.31 4.69 5.41 4.36 4.74 5.46 4.33 4.71 5.42 4.34 4.72 5.43 4.33 4.70 5.42 4.34 4.72 5.43 See www.treas.gov/domfin/statistics.htm Valuation Relationship Between Short and Long Interest Rates I. Expectations Hypothesis (with certainty) ( 1 + tRN) = [(1 + tR1)(1 + t+1r1)(1 + t+2r1)(1 + t+N-1r1)]

1/N Where R = actual market interest rate r = expected interest rate pre-subscript is the time period for which rates are applicable post-subscript is the maturity of the bond so tR1 is todays (period t) one year actual yield (e.g. 2.21%) and t+1r1 is the one-year expected yield one year from today. Maturity 1 2 3 FV of $1 1.02210 1.05678 1.10166 Implied 1-Year Yield 2.2100 3.3934 4.2467 Rate change (bp) 118bp 85bp Daily Treasury Yield Curve Rates Historical Data November 2004 Date 1 mo 3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 11/01/04 1.79 1.99 2.20 2.34

2.61 2.89 3.36 3.76 4.11 4.84 11/02/04 1.86 1.97 2.19 2.33 2.60 2.86 3.34 3.75 4.10 4.84 11/03/04 1.83 1.96 2.18 2.32 2.60 2.85 3.35 3.74 4.09 4.83 11/04/04 1.85 1.98 2.19

2.34 2.63 2.89 3.37 3.76 4.10 4.82 11/05/04 1.86 2.03 2.27 2.44 2.80 3.04 3.51 3.88 4.21 4.92 11/08/04 1.88 2.07 2.30 2.47 2.80 3.08 3.51 3.88 4.22 4.95 * 30-year Treasury constant maturity series was discontinued as of 2/18/02. See Long-Term Average Rate for more information. Source: http://www.treas.gov/offices/domestic-finance/debt-management/interestrate/yield.html Valuation

A Generic Interest Rate Model Change in interest rate = drift parameter * small time interval + volatility parameter * random shock Or, in the language of Calculus, dr = drift * dt + volatility * dz where dr = (very short term) change in interest rates drift parameter = a (rLT - rt) where 0 < a < 1 (mean reversion) dt = change in time period volatility parameter, = standard deviation of interest rates random shock = Brownian motion The drift and volatility parameters need not be constantsthey can depend on time period and/or interest rates. Valuation Brownian Motion Normally distributed (e.g. bell shaped) The distribution is centered at zero Variance scaled to one Draws are serially independent Valuation Some Interest Rate Models Vasicek: dr = a(rLT r)dt + dz Hull-White: dr = a(t)((t) r)dt + (t)dz Cox-Ingersoll-Ross: dr = a(t)((t) r)dt + (t)rdzrdz There are others.

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